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Relaxation, frequency shifts and other phenomena at the transition between diffusion and ballistic motion

There are many fields where the transition from diffusive to ballistic motion is important. Here we deal with relaxation processes in nmr in gases. Correlation functions for trajectory variables (position and velocity) valid across this transition are known for several geometries in the case of specular wall reflections. In this work we show that the conditional probability density p(r,t|r_0,t_0) for a random walk satisfies the telegrapher's equation and present an analytic solution for this function. We will use it for calculating the relaxation due to an axion mediated force and a magnetic dipole impurity.